By Ivana Kovacic, Michael J. Brennan

The Duffing Equation: Nonlinear Oscillators and their Behaviour brings jointly the result of a wealth of disseminated learn literature at the Duffing equation, a key engineering version with an unlimited variety of purposes in technology and engineering, summarizing the findings of this learn. each one bankruptcy is written via knowledgeable contributor within the box of nonlinear dynamics and addresses a special kind of the equation, touching on it to numerous oscillatory difficulties and obviously linking the matter with the maths that describe it. The editors and the participants clarify the mathematical ideas required to review nonlinear dynamics, supporting the reader with little mathematical history to appreciate the text.The Duffing Equation offers a reference textual content for postgraduate and scholars and researchers of mechanical engineering and vibration / nonlinear dynamics in addition to a useful gizmo for practicing mechanical engineers.Includes a bankruptcy dedicated to old historical past on Georg Duffing and the equation that was once named after him.Includes a bankruptcy exclusively dedicated to functional examples of structures whose dynamic behaviour is defined through the Duffing equation.Contains a accomplished therapy of a number of the varieties of the Duffing equation.Uses experimental, analytical and numerical equipment in addition to innovations of nonlinear dynamics to regard the actual structures in a unified means.

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**Example text**

When connected to a mass, this system can potentially have very complex dynamics because of the snap-through behaviour. This is discussed fully in Chapter 7. When d~ < 1, the gradient is always positive, so the linear stiffness is always positive. It can also be seen that as the displacement increases then the stiffness increases as expected due to the hardening characteristic of the spring as discussed above. 4: (a) Nondimensional force–deflection characteristic as the original length of each spring is varied; (b) Nondimensional potential energy as a function of nondimensional displacement as the original length of each springs is varied.

In this paper he described several nonlinear engineering problems, one of which involved subharmonic resonance due to nonlinear restoring forces. Duffing’s book was listed in the bibliography. In the late 1930s and 1940s a group of applied mathematicians worked on nonlinear problems in New York University. These were led by Richard Courant, who left Germany in the mid-1930s, where he had been an assistant to Hilbert at G€ottingen. His group included Kurt Friedrichs, his former student who left Germany in 1937 to join him, and James Stoker, who subsequently wrote the seminal book on nonlinear vibrations [22].

Note the quasi-zero stiffness characteristic of the isolators at a certain deflection. ) THE DUFFING EQUATION 37 the system can change considerably. To illustrate the effects that a mistuned system can have on the equation of motion a constant force F0 is applied to the system so that ~€ y þ g~y3 ¼ F~0 þ F~ cos O ~t ð2:7:6Þ ^ 0 ¼ F0 =kv ys . The deviation of the mass from the tuned static equilibrium where F position is equal to this force divided by the weight of the suspended mass. This detuning could also occur because of the steady acceleration of an aircraft or a vehicle, for example, in which the isolation system is situated.